Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach)
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914585737428992 |
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| author | Hong, Hoon Profili, Daniel Sendra, J. Rafael |
| author_facet | Hong, Hoon Profili, Daniel Sendra, J. Rafael |
| contents | Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that their eigenvalues have the given configuration. We give an algorithm which expresses the eigenvalue configuration problem as a real root counting problem of certain symmetric polynomials, whose roots can be counted using the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00089 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach) Hong, Hoon Profili, Daniel Sendra, J. Rafael Algebraic Geometry Symbolic Computation Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that their eigenvalues have the given configuration. We give an algorithm which expresses the eigenvalue configuration problem as a real root counting problem of certain symmetric polynomials, whose roots can be counted using the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs. |
| title | Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach) |
| topic | Algebraic Geometry Symbolic Computation |
| url | https://arxiv.org/abs/2401.00089 |